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=====1.3.4.2.  Sign Relations : A Primer=====

=====1.3.4.2.  Sign Relations : A Primer=====

To the extent that their structures and functions can be discussed at all, it is likely that all of the formal entities that are destined to develop in this approach to inquiry will be instances of a class of three-place relations called "sign relations".  At any rate, all of the formal structures that I have examined so far in this area have turned out to be easily converted to or ultimately grounded in sign relations.  This class of triadic relations constitutes the main study of the "pragmatic theory of signs", a branch of logical philosophy devoted to understanding all types of symbolic representation and communication.

To the extent that their structures and functions can be discussed at all, it is likely that all of the formal entities that are destined to develop in this approach to inquiry will be instances of a class of [[triadic relation|three-place relation]]s called ''[[sign relation]]s''.  At any rate, all of the formal structures that I have examined so far in this area have turned out to be easily converted to or ultimately grounded in sign relations.  This class of triadic relations constitutes the main study of the ''pragmatic theory of signs'', a branch of logical philosophy devoted to understanding all types of symbolic representation and communication.

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey).  Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation.  Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey).  Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation.  Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations.  Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs.

Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations.  Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns:  "Ann", "Bob", "I", "you".

Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns:  "Ann", "Bob", "I", "you".

In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use.  The "system of interpretation" (SOI) associated with each language user can be represented in the form of an individual three-place relation called the ''[[sign relation]]'' of that interpreter.

In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use.  The "system of interpretation" (SOI) associated with each language user can be represented in the form of an individual three-place relation called the ''[[sign relation]]'' of that interpreter.

Understood in terms of its set-theoretic extension, a sign relation ''L'' is a subset of a cartesian product ''O''×''S''×''I''.  Here, ''O, ''S'', and ''I'' are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation ''L'' ⊆ ''O''×''S''×''I''.  In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having ''I'' ⊆ ''S''.  In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a ''syntactic domain''.  In the forthcoming examples, ''S'' and ''I'' are identical as sets, so the very same elements manifest themselves in two distinct roles of the sign relations in question.  When it is necessary to refer to the whole set of objects and signs in the union of the domains ''O'', ''S'', and ''I'' for a given sign relation ''L'', one may call this the "world of ''L''" and write ''W'' = ''W''(''L'') = ''O'' ∪ ''S'' ∪ ''I''.

Understood in terms of its set-theoretic extension, a sign relation ''L'' is a subset of a cartesian product ''O''×''S''×''I''.  Here, ''O'', ''S'', and ''I'' are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation ''L'' ⊆ ''O''×''S''×''I''.  In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having ''I'' ⊆ ''S''.  In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a ''syntactic domain''.  In the forthcoming examples, ''S'' and ''I'' are identical as sets, so the very same elements manifest themselves in two distinct roles of the sign relations in question.  When it is necessary to refer to the whole set of objects and signs in the union of the domains ''O'', ''S'', and ''I'' for a given sign relation ''L'', one may call this the "world of ''L''" and write ''W'' = ''W''(''L'') = ''O'' ∪ ''S'' ∪ ''I''.

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible when the examples become more complicated, I introduce the following abbreviations:

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible when the examples become more complicated, I introduce the following abbreviations: